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polygon

1 Definitions

We follow Forder [2] for most of this entry.
The termpolygon can be defined if one has a definition of an interval. For this
entry we use betweenness geometry. A betweenness geometry
is just one for which there is a set of points and a betweenness relationBBB defined.
Rather than write (a,b,c)∈BabcB(a,b,c)\in B we write a*b*cabca*b*c.

1.

If aaa and bbb are distinct points, the linea⁢babab is the set of
all points ppp such that p*a*bpabp*a*b or a*p*bapba*p*b or a*b*pabpa*b*p. It can be shown
that the line a⁢babab and the line b⁢ababa are the same set of points.

2.

If ooo and aaa are distinct points, a ray[oafragmentsnormal-[oa[oa is the set of all points ppp such that
p=opop=o or o*p*aopao*p*a or o*a*poapo*a*p.

3.

If aaa and bbb are distinct points, the open interval is the set of points
ppp such that a*p*bapba*p*b. It is denoted by (a,b).ab(a,b).

4.

If aaa and bbb are distinct points, the closed interval is
(a,b)∪{a}∪{b}abab(a,b)\cup\{a\}\cup\{b\}, and denoted by [a,b].ab[a,b].

5.

The waya1⁢a2⁢…⁢ansubscripta1subscripta2normal-…subscriptana_{1}a_{2}\ldots a_{n} is the finite set of points {a1,…,an}subscripta1normal-…subscriptan\{a_{1},\ldots,a_{n}\}
along with the open intervals (a1,a2),(a2,a3),…,(an-1,an)subscripta1subscripta2subscripta2subscripta3normal-…subscriptan1subscriptan(a_{1},a_{2}),(a_{2},a_{3}),\ldots,(a_{{n-1}},a_{n}).
The points a1,…,ansubscripta1normal-…subscriptana_{1},\ldots,a_{n} are called the vertices of the way, and the
open intervals are called the sides of the way.
A way is also called a broken line.
The closed intervals [a1,a2],…,[an-1,an]subscripta1subscripta2normal-…subscriptan1subscriptan[a_{1},a_{2}],\ldots,[a_{{n-1}},a_{n}] are called the side-intervals of
the way. The lines a1⁢a2,…,an-1⁢ansubscripta1subscripta2normal-…subscriptan1subscriptana_{1}a_{2},\ldots,a_{{n-1}}a_{n} are called the side-lines
of the way.
The way a1⁢a2⁢…⁢ansubscripta1subscripta2normal-…subscriptana_{1}a_{2}\ldots a_{n} is said to joina1subscripta1a_{1} to ansubscriptana_{n}.
It is assumed that ai-1,ai,ai+1subscriptai1subscriptaisubscriptai1a_{{i-1}},a_{i},a_{{i+1}} are not collinear.

6.

A way is said to be simple if it does not meet itself. To be precise,
(i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals
meet in any point.

7.

A polygon is a way a1⁢a2⁢…⁢ansubscripta1subscripta2normal-…subscriptana_{1}a_{2}\ldots a_{n} for which a1=ansubscripta1subscriptana_{1}=a_{n}. Notice that there is
no assumption that the points are coplanar.

A region is a set of points not all collinear, any two of which can be joined by points of a way using
only points of the region.

10.

A region RRR is convex if for each pair of points a,b∈RabRa,b\in R the open interval (a,b)ab(a,b) is
contained in R.RR.

11.

Let XXX and YYY be two sets of points. If there is a set of points SSS such that every way
joining a point of XXX to a point of YYY meets SSS then SSS is said to separateXXX from YYY.

12.

If a1⁢a2⁢…⁢ansubscripta1subscripta2normal-…subscriptana_{1}a_{2}\ldots a_{n} is a polygon, then the angles of the polygon are
∠⁢an⁢a1⁢a2,∠⁢a1⁢a2⁢a3normal-∠subscriptansubscripta1subscripta2normal-∠subscripta1subscripta2subscripta3\angle a_{n}a_{1}a_{2},\angle a_{1}a_{2}a_{3}, and so on.

Now assume that all points of the geometry are in one plane. Let PPP be a polygon. (PPP is called
a plane polygon.)

1.

A ray or line which does not go through a vertex of PPP will be called suitable.

2.

An inside pointaaa of PPP is one for which a suitable ray from aaa
meets PPP an odd number of times. Points that are not on or inside PPP are said to be outsidePPP.

3.

Let {Pi}subscriptPi\{P_{i}\} be a set of polygons. We say that {Pi}subscriptPi\{P_{i}\}dissectPPP if the following
three conditions are satisfied: (i) PisubscriptPiP_{i} and PjsubscriptPjP_{j} do not have a common inside point for i≠jiji\not=j,
(ii) each inside point of PPP is inside or on some PisubscriptPiP_{i} and (iii) each inside point of PisubscriptPiP_{i} is
inside PPP.

4.

A convex polygon is one whose inside points are all on the same side of any side-line
of the polygon.

2 Theorems

Assume that all points are in one plane. Let PPP be a polygon.

1.

It can be shown that PPP separates the other points of the plane into at least two regions and that
if PPP is simple there are exactly two regions. Moise proves this directly in [3], pp. 16-18.

2.

It can be shown that PPP can be dissected into triangles{Ti}subscriptTi\{T_{i}\} such that
every vertex of a TisubscriptTiT_{i} is a vertex of PPP.

3.

The following theorem of Euler can be shown: Suppose PPP is dissected into f>1f1f>1 polygons
and that the total number of vertices of these polygons is vvv, and the number of open intervals
which are sides is eee. Then

v-e+f=1vef1v-e+f=1

.

A plane simple polygon with nnn sides is called an nnn-gon, although for small nnn
there are more traditional names:

Mathematics Subject Classification

Comments

Although Mathprof has adopted and edited this entry so that the concepts are mathematically precise, I am concerned about the accessibility of this entry to non-mathematicians. The term "polygon" is one that people encounter very early on and thus, in my opinion, should have an entry that gives basic information about polygons that are both mathematically precise and accessible to the general population.

Also, I am pretty sure that this entry went up for adoption due to the fact that terms such as "interior angle" and "exterior angle" are difficult to define in a mathematically precise way. Nevertheless, these terms (along with "angle sum") are commonly used and should appear somewhere in PM.

I have been toying with creating another polygon entry which is meant for people who do not have the mathematical background that is necessary to understand the bulk of the content of the current entry. Before doing this, I wanted to get other people's opinions on this matter.

While I think it's a good idea for there to be another entry on polygon which anyone with a high school education can understand, it wouldn't hurt for this entry to have a diagram or two. In fact, it would be nice for the two entries to have the same pictures.

As for the canonical name, you could do what Anton did the simple entry about length.

Sorry for taking so long to get back to this, but I plan on finally adding a polygon entry as described in previous posts some time this weekend. Of course, as I want it to be an entry that virtually anyone can read, there will be pictures (pretty pictures I hope!).